If $s$ is transcendental, is $\frac{1}{s}$ transcendental?

Question

From Wikipedia, the number $3.300330000000000330033\dots$ and its reciprocal $0.30300000303\dots$ are both transcendental. I am wondering:

If $s$ is transcendental, is $\dfrac 1s$ necessarily transcendental?

I am assuming the answer is not known, with transcendental numbers being complicated and whatnot; if that is the case, is much known about the reciprocal of transcendental numbers? Otherwise, if my suspicion is false, is there a known transcendental number whose reciprocal is a counterexample?

Answer 1

The answer is yes: If $1/s$ is algebraic, then $p(1/s) = 0$ for some nontrivial polynomial $$p(x) = a_0 x^0 + \cdots + a_n x^n$$ with rational coefficients. But then $$0 = s^n p(1/s) = a_0 s^n + \cdots + a_n s^0,$$ which implies that $s$ is algebraic.

Answer 2

The algebraic numbers form a field. Thus, if $1/s$ were algebraic, then $1/(1/s)=s$ would also be algebraic, a contradiction, showing that $1/s$ must be transcendental if $s$ is.